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arxiv: 2604.05214 · v1 · submitted 2026-04-06 · 🧮 math.LO

On 2-generated minimal Taylor algebras of size 4

Zarathustra Brady , Petar {\DJ}api\'c , Vuka\v{s}in {\DJ}inovi\'c , Petar Markovi\'c , Aleksandar Proki\'c , Veljko Tolji\'c , Vlado Uljarevi\'c This is my paper

Pith reviewed 2026-05-10 18:37 UTC · model grok-4.3

classification 🧮 math.LO
keywords minimal Taylor algebras2-generated algebrasuniversal algebracongruence latticesclassificationterm equivalencefour-element domains
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The pith

Any 2-generated minimal Taylor algebra on a four-element domain is not simple.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that every 2-generated minimal Taylor algebra with a four-element domain fails to be simple, meaning each such algebra admits at least one non-trivial congruence. The authors complete the argument by enumerating all algebras of this type up to isomorphism and term-equivalence. A reader interested in the structure of Taylor algebras would see the result as a concrete restriction on the possible congruence lattices and term sets that can arise under these generation and size constraints. The classification supplies explicit representatives that can be checked directly against the definition of minimality and the Taylor property.

Core claim

Any 2-generated minimal Taylor algebra on a domain of size 4 is not simple. In addition, all such algebras are found up to isomorphism and term-equivalence.

What carries the argument

The congruence lattice of a 2-generated minimal Taylor algebra on four elements, together with the term operations that enforce the Taylor condition.

If this is right

  • Every such algebra possesses a proper nontrivial congruence.
  • The set of all such algebras is finite and can be listed by direct computation of possible term tables.
  • Term-equivalence classes correspond to distinct ways the four elements can interact under the Taylor terms while remaining 2-generated.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The result supplies a base case that may be used to test whether similar non-simplicity holds for 2-generated minimal Taylor algebras on larger domains.
  • The explicit list allows one to verify whether the non-simplicity persists after adding constants or changing the generating set.
  • The classification isolates the precise term conditions that force the existence of a congruence on four elements.

Load-bearing premise

The algebra satisfies the definition of a minimal Taylor algebra and is generated by exactly two elements on a four-element domain.

What would settle it

Exhibiting one concrete 2-generated minimal Taylor algebra on four elements whose congruence lattice is trivial would refute the main claim.

read the original abstract

We prove that any 2-generated minimal Taylor algebra on a domain of size 4 is not simple. In addition, we find all such algebras up to isomorphism and term-equivalence.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript proves that every 2-generated minimal Taylor algebra on a 4-element domain is not simple. It additionally enumerates all such algebras up to isomorphism and term-equivalence via an exhaustive case analysis of possible Taylor terms, generated clones, and congruence lattices.

Significance. The explicit classification supplies a complete list for the smallest non-trivial domain size in this setting, which can serve as a foundation for further results on simplicity, subalgebras, and CSP complexity in universal algebra. The non-simplicity claim follows directly from the enumeration.

major comments (1)
  1. [§4] §4 (Case analysis for 2-generated clones): the completeness of the list in Theorem 5.1 rests on the manual enumeration of all possible minimal Taylor terms and the clones they generate. The argument does not include an explicit enumeration of all 2-generated clones containing a Taylor operation or a proof that no additional cases exist outside the considered congruence lattices; a missed simple algebra would falsify both the non-simplicity theorem and the classification claim.
minor comments (2)
  1. [§2] Notation for term-equivalence classes is introduced without a dedicated preliminary subsection; a short table listing representatives and their term operations would improve readability.
  2. [Abstract] The abstract states the two main results but does not indicate that the proof proceeds by exhaustive classification; adding one sentence on the method would help readers assess the scope.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting the need to strengthen the justification of completeness in the case analysis. We address this point below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [§4] §4 (Case analysis for 2-generated clones): the completeness of the list in Theorem 5.1 rests on the manual enumeration of all possible minimal Taylor terms and the clones they generate. The argument does not include an explicit enumeration of all 2-generated clones containing a Taylor operation or a proof that no additional cases exist outside the considered congruence lattices; a missed simple algebra would falsify both the non-simplicity theorem and the classification claim.

    Authors: We agree that the current presentation of the case analysis in §4 would benefit from an explicit argument establishing that the considered cases are exhaustive. In the revised manuscript we will add a short preliminary subsection to §4 that begins by classifying all possible minimal Taylor terms on a 4-element domain up to the action of the symmetric group. Because any Taylor term is idempotent and satisfies the Taylor condition, only a small number of inequivalent operations arise (majority, minority, and the remaining low-arity Taylor operations); we list them explicitly together with the equations they satisfy. For each such term we then describe the 2-generated clone it generates and the possible congruence lattices that can appear in a 2-generated algebra containing it. We prove that these lattices exhaust all possibilities by observing that, in a 2-generated algebra, every congruence is determined by the pairs of elements that are identified under the generators, and the Taylor identities force the existence of a non-trivial congruence in every case. This added argument will make clear that no simple algebra has been overlooked. revision: yes

Circularity Check

0 steps flagged

No circularity: direct classification proof on finite domain

full rationale

The paper states a standard mathematical result: a proof that every 2-generated minimal Taylor algebra of size 4 is not simple, together with an explicit list of all such algebras up to isomorphism and term equivalence. The derivation relies on the definitions of Taylor algebras, minimality, 2-generation, and standard universal-algebraic tools (congruences, term operations). No self-definitional equations, fitted parameters renamed as predictions, load-bearing self-citations, or imported uniqueness theorems appear in the provided abstract or description. The finite domain makes exhaustive case analysis feasible in principle; the claim is therefore self-contained and does not reduce to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard definitions and properties of Taylor algebras, minimality, simplicity, and term-equivalence from universal algebra literature; no new free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Standard definitions and properties of Taylor algebras, minimal Taylor algebras, congruences, and term operations in universal algebra
    The result builds directly on established concepts in the field without new axioms.

pith-pipeline@v0.9.0 · 5355 in / 1116 out tokens · 35754 ms · 2026-05-10T18:37:28.670638+00:00 · methodology

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Reference graph

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14 extracted references · 14 canonical work pages

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